Division of Rational Functions

Definition

The division of rational functions refers to the process of dividing one rational function by another, resulting in a new rational function. This operation is essential in algebra as it allows for simplification and solving complex expressions. Understanding how to divide rational functions involves grasping polynomial long division or synthetic division when necessary, as well as recognizing the potential for simplifying the resulting function by factoring out common terms.

5 Must Know Facts For Your Next Test

1. To divide rational functions, express both the numerator and the denominator as their simplest forms first, if possible.
2. When dividing a rational function by another, if the degree of the numerator is less than the degree of the denominator, the result is a proper fraction.
3. If both numerator and denominator share common factors, it's crucial to simplify before performing the division to avoid complexity.
4. The remainder obtained during polynomial long division can be expressed as a fraction added to the quotient, giving insight into how closely the two functions approximate each other.
5. Understanding how to identify and handle vertical asymptotes is key when dividing rational functions, as they indicate points of discontinuity.

Review Questions

• How can you determine whether to use polynomial long division or synthetic division when dividing rational functions?
  • The choice between polynomial long division and synthetic division generally depends on the degree and form of the polynomial being divided. Polynomial long division is applicable for any polynomial, while synthetic division simplifies the process for linear divisors. If the divisor is a linear polynomial in the form $$x - c$$, synthetic division is often quicker and more efficient.
• Explain how common factors in rational functions affect the outcome of their division.
  • Common factors between the numerator and denominator must be identified and canceled before performing the division. This cancellation simplifies the process and ensures that any undefined behavior at certain values does not distort results. Failing to simplify can lead to incorrect conclusions about behavior near vertical asymptotes or discontinuities.
• Evaluate how dividing rational functions contributes to understanding their graphs, especially concerning asymptotic behavior.
  • Dividing rational functions provides insights into their graphs by revealing vertical and horizontal asymptotes, which indicate critical behavior as x approaches specific values. The division helps illustrate how different degrees of polynomials in numerator and denominator influence these asymptotes. Moreover, analyzing remainders in long division can highlight behavior as x goes to infinity, deepening comprehension of the overall function's trends and limits.

Related terms

Rational Function: A rational function is a function expressed as the quotient of two polynomials, typically in the form $$rac{P(x)}{Q(x)}$$ where both P and Q are polynomials.

Polynomial Long Division: A method used to divide polynomials similar to traditional long division, often used when dividing a polynomial by another polynomial.

Vertical Asymptote: A vertical line that corresponds to values of x that make the denominator of a rational function equal to zero, indicating points where the function is undefined.